Graphs of Linear Equations | 线性方程的图像

📚 Graphs of Linear Equations | 线性方程的图像

In this topic, you will learn how to plot and interpret straight-line graphs on the coordinate plane. We begin by reviewing the Cartesian grid, then move through constructing tables of values, plotting points, and drawing lines. You will discover how the equation y = mx + c controls the slope and position of a line, and you will practise using graphs to solve simple problems. By the end, you should feel confident reading and drawing linear graphs accurately for your Cambridge KS3 mathematics course.

在本主题中，你将学习如何在坐标平面上绘制并解读直线图像。我们先回顾直角坐标网格，然后逐步学习构建数值表、描点并连线。你将发现方程 y = mx + c 如何控制直线的斜率和位置，并练习用图像解决简单问题。学完本节后，你应当能够自信地准确阅读和绘制线性图像，为剑桥 KS3 数学课程打下基础。

1. Coordinates and the Cartesian Plane | 坐标与直角坐标系

A coordinate plane is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. The point where they cross is called the origin, written as (0, 0). Every position on the grid can be described by an ordered pair (x, y), where x tells you how far to move horizontally and y tells you how far to move vertically.

坐标平面由两条互相垂直的数轴构成：水平的 x 轴和竖直的 y 轴。它们的交点称为原点，记作 (0, 0)。网格上的每一个位置都可以用一个有序对 (x, y) 来描述，其中 x 表示水平移动的距离，y 表示竖直移动的距离。

When writing coordinates, the x-value always comes first. For example, (3, 2) means move 3 units right from the origin along the x-axis, then 2 units up. If a number is negative, move left or down instead.

写坐标时，x 值始终在前。例如 (3, 2) 表示从原点沿 x 轴向右移动 3 个单位，再向上移动 2 个单位。如果数字为负，则向左或向下移动。

2. What is a Linear Equation? | 什么是线性方程？

A linear equation is an equation whose graph forms a straight line. When we talk about linear graphs, we usually work with equations that can be written in the form y = mx + c, where m and c are numbers. For KS3, m is often called the gradient or slope, and c is the y-intercept — the place where the line crosses the y-axis.

线性方程是指其图像是一条直线的方程。我们谈论线性图像时，通常使用可以写成 y = mx + c 形式的方程，其中 m 和 c 都是数字。在 KS3 阶段，m 常被称为斜率或坡度，c 是 y 轴截距——即直线与 y 轴相交的位置。

For example, y = 2x + 1 is a linear equation. The slope is 2, and the y-intercept is 1. The graph of this equation is a straight line that rises as you move to the right. Another example, y = –x + 4, has a negative slope and crosses the y-axis at 4.

例如 y = 2x + 1 就是一个线性方程。斜率为 2，y 轴截距为 1。该方程的图像是一条向右逐渐上升的直线。再比如 y = –x + 4，斜率为负，与 y 轴交于 4 的位置。

3. Creating a Table of Values | 建立数值表

To draw a linear graph, you first need a set of coordinates that satisfy the equation. A table of values helps you organise these points. Choose at least three x-values, usually –2, 0, and 2, or –1, 0, and 1. Substitute each x into the equation to find the matching y-value.

要绘制线性图像，你首先需要找到一组满足方程的点坐标。数值表可以帮助你整理这些点。至少选择三个 x 值，通常取 –2、0、2 或 –1、0、1。将每个 x 值代入方程，求出对应的 y 值。

For y = 2x + 1:

• x = –2 → y = 2(–2) + 1 = –4 + 1 = –3 → (–2, –3)
• x = 0 → y = 2(0) + 1 = 0 + 1 = 1 → (0, 1)
• x = 2 → y = 2(2) + 1 = 4 + 1 = 5 → (2, 5)

对于 y = 2x + 1：

• x = –2 → y = 2(–2) + 1 = –4 + 1 = –3 → (–2, –3)
• x = 0 → y = 2(0) + 1 = 0 + 1 = 1 → (0, 1)
• x = 2 → y = 2(2) + 1 = 4 + 1 = 5 → (2, 5)

4. Plotting Points on a Grid | 在网格上描点

Once you have your coordinates, plot each point carefully on graph paper or a coordinate grid. Use a cross or dot to mark the position, and label the point if needed. Accuracy is important: a point that is even slightly off will affect how straight your line looks.

得到坐标后，在方格纸或坐标网格上仔细描出每个点。用叉号或圆点标记位置，必要时可标注坐标。准确性很重要：即便轻微偏离，也会影响直线的笔直程度。

For the points (–2, –3), (0, 1), and (2, 5), move left from the origin for negative x and down for negative y. The more points you plot, the easier it is to see if a mistake has been made.

对于点 (–2, –3)、(0, 1) 和 (2, 5)，当 x 为负时从原点向左移动，y 为负时向下移动。描出的点越多，就越容易发现是否存在错误。

5. Drawing the Straight Line | 绘制直线

After plotting the points, use a ruler to draw a single straight line that passes through all of them. The line should extend beyond the first and last points to show that it continues in both directions. Add arrows at both ends to indicate that the line goes on forever.

描出所有点后，用直尺画一条穿过所有点的直线。直线应当延伸到首尾两点之外，以表明它在两个方向上延续。在两端添加箭头，表示直线无限延伸。

If your line does not go through every point smoothly, check your calculations and point plotting. A third or fourth point can confirm whether the line is correct.

如果直线没有平滑地穿过每一个点，请检查计算和描点过程。增加第三个或第四个点可以验证直线是否正确。

6. Understanding Slope and Intercept | 理解斜率和截距

The slope m of the line y = mx + c tells you how steep the line is. It is calculated by dividing the vertical change (rise) by the horizontal change (run) between any two points on the line. A larger m means a steeper line.

直线 y = mx + c 的斜率 m 告诉你线有多陡。它可以用直线上任意两点之间的竖直变化量（高度差）除以水平变化量（水平距离）来计算。m 越大，线越陡。

The y-intercept c is simply the y-coordinate where the line crosses the y-axis. In y = 2x + 1, the line crosses at (0, 1). In y = –3x + 5, the line crosses at (0, 5). This is one of the easiest points to plot.

y 轴截距 c 就是直线与 y 轴相交点的 y 坐标。对于 y = 2x + 1，直线交于 (0, 1)；对于 y = –3x + 5，直线交于 (0, 5)。这是最容易描出的点之一。

7. Positive and Negative Slopes | 正斜率和负斜率

When m is positive, the line slopes upwards from left to right. For example, y = 3x – 2 goes up as x increases. When m is negative, the line slopes downwards from left to right, like y = –2x + 4. If m = 0, the line is horizontal.

当 m 为正时，直线从左向右上升。例如 y = 3x – 2 随 x 增加而上升。当 m 为负时，直线从左向右下降，比如 y = –2x + 4。如果 m = 0，直线是水平的。

You can check the sign of the slope by looking at the change in y as x increases by 1. If y gets bigger, the slope is positive; if it gets smaller, the slope is negative. This helps you predict the shape of the graph before plotting.

你可以通过观察 x 增加 1 时 y 的变化来判断斜率的正负。y 变大，则斜率为正；y 变小，则斜率为负。这有助于在描画之前预测图像的形状。

8. Horizontal and Vertical Lines | 水平和垂直线

Not all straight lines can be written as y = mx + c. A horizontal line has an equation like y = 3. Here, every point has the same y-coordinate, so the line is flat and parallel to the x-axis. Its slope is zero.

并非所有直线都能写成 y = mx + c 的形式。水平线的方程形如 y = 3。此时每个点的 y 坐标都相同，因此直线是水平的，平行于 x 轴。其斜率为零。

A vertical line has an equation like x = –2. Every point has the same x-coordinate, so the line goes straight up and down. The slope of a vertical line is undefined because you cannot divide by zero (run = 0).

垂直线的方程形如 x = –2。每个点的 x 坐标都相同，因此直线竖直上下。垂直线的斜率是未定义的，因为你不能除以零（水平距离为零）。

9. Using Graphs to Solve Equations | 利用图像解方程

Linear graphs can help you solve simple equations visually. To solve 2x + 1 = 5, you can draw the line y = 2x + 1 and the horizontal line y = 5 on the same grid. The x-coordinate of their intersection is the solution.

线性图像可以帮助你可视化地解简单方程。要解 2x + 1 = 5，你可以在同一网格上画出直线 y = 2x + 1 和水平线 y = 5。它们交点的 x 坐标就是方程的解。

In this case, the lines meet at (2, 5), so the solution is x = 2. This method also works for simultaneous equations if you plot two linear equations and find where they cross.

在这个例子中，两直线交于点 (2, 5)，因此解为 x = 2。这个方法同样适用于联立方程，只需画出两条直线并找出它们的交点即可。

10. Common Mistakes and Tips | 常见错误与提示

A frequent error is swapping the x and y when plotting. Always move horizontally first, then vertically. Another mistake is misreading negative signs, for example thinking (–2, –3) is (2, 3). Double-check your substitution carefully.

一个常见错误是在描点时把 x 和 y 弄反。请始终先水平移动，再竖直移动。另一个错误是误读负号，例如把 (–2, –3) 想成 (2, 3)。务必仔细检查代入过程。

When drawing the line, make sure your ruler is aligned with all points. A line that misses a point usually means an arithmetic error. Always use at least three points to be safe. A table with a zero x-value often makes plotting easier.

画直线时，请确保直尺对准所有点。如果直线没穿过某个点，通常说明计算有误。为保险起见，至少使用三个点。在表格中包含 x = 0 往往能使描图变得更简单。

11. Real-life Applications | 实际应用

Linear graphs appear in many everyday situations, such as converting currencies, calculating mobile phone costs with a fixed monthly charge, or finding the distance travelled at constant speed. The slope represents a rate, and the intercept often stands for a fixed starting amount.

线性图像出现在许多日常情境中，例如货币兑换、计算带有固定月费的手机费用，或求匀速行驶的距离。斜率代表变化率，截距通常表示固定的起始量。

For instance, a taxi fare model F = 2d + 3 has a slope of 2 (cost per kilometre) and a y-intercept of 3 (flag-down fee). Plotting this helps you quickly estimate costs for different distances.

例如出租车计费模型 F = 2d + 3，斜率为 2（每公里费用），y 轴截距为 3（起步价）。画出图像能帮助你快速估算不同距离的费用。

12. Practice Question Walkthrough | 练习题讲解

Question: Draw the graph of y = 4 – x for integer x-values from –2 to 3. Find where the line crosses the x-axis.

题目：画出 y = 4 – x 的图像，x 取从 –2 到 3 的整数。找出直线与 x 轴的交点。

First, create a table: x = –2 → y = 4 – (–2) = 6; x = –1 → 5; x = 0 → 4; x = 1 → 3; x = 2 → 2; x = 3 → 1. Plot these coordinates. The line slopes downward because the slope is –1. It crosses the x-axis when y = 0. Set 4 – x = 0, so x = 4. Even though 4 is not in the table, the line extends to (4, 0).

首先，建立表格：x = –2 → y = 4 – (–2) = 6；x = –1 → 5；x = 0 → 4；x = 1 → 3；x = 2 → 2；x = 3 → 1。描出这些点。由于斜率为 –1，直线向下倾斜。它与 x 轴交于 y = 0 的位置。设 4 – x = 0，解得 x = 4。虽然 4 不在表格中，但直线延伸后经过 (4, 0)。

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